Fibonacci proof by induction

1. Dec 7, 2008

kathrynag

1. The problem statement, all variables and given/known data
$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

2. Relevant equations

3. The attempt at a solution
P(k+1):$$F_{2k-1}+F_{2k+1}$$=$$F_{2k+2}$$

2. Dec 7, 2008

mutton

What happened to F_1 and F_3?

3. Dec 7, 2008

kathrynag

well, it's F_1+F_3+....+F_2n-1

4. Dec 7, 2008

mutton

Then P(k + 1) needs to be changed accordingly.

What have you tried so far?

5. Dec 7, 2008

kathrynag

I've tried plugging in numbers.

6. Dec 7, 2008

mutton

And how did that work out?

The definition of Fibonacci numbers will be helpful in the induction proof.

7. Dec 8, 2008

kathrynag

If I plug in 1, I just get F_1, so 1=1
If I plug in 2, I get F_3, so 1+2=F_4, 3=3

8. Dec 8, 2008

kathrynag

F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1

Now I'm stumped...

9. Dec 8, 2008

HallsofIvy

Staff Emeritus
What was "F_1+ F_3+ ...+ F_2n-1"?

In your first post you said the problem was to prove that
$$F_{1}+F_{3}+F_{2n-1}$$=$$F_{2n}$$

Are you saying now it is actually to prove that
$$F_{1}+F_{3}+\cdot\cdot\cdot +F_{2n-1}$$=$$F_{2n}$$?

10. Dec 8, 2008

mutton

Very close. What happens when 2 consecutive Fibonacci numbers are added?

11. Dec 8, 2008

kathrynag

It equals the 3rd Fibonnacci number.
F_1+F-2=F_3

so F_2k+F_2k+1=F_2k+1+1

12. Dec 8, 2008

mutton

And that's exactly what you wanted to show.

13. Dec 8, 2008

kathrynag

Ok thanks!

14. Dec 9, 2008

epenguin

Sorry but maybe the problem is not properly stated? Are the F defined to be Fibonacci numbers?

Then F2n = F2n-1 + F2n-2

So if you are then asking also that

F2n = F2n-1 + F1 + F3

then

F2n-2 = F1 + F3

which is not making much sense.